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\begin{document}

\title{Modeller Kuram\i\ (MAT 414)\\
Ara S\i nav\i}
\date{6 Nisan 2014}
\author{David Pierce}
%\abstract{}
\maketitle
\thispagestyle{empty}

\begin{problem}
$T_<{}^*$, u\c csuz yo\u gun do\u grusal s\i ralamalar teorisi olsun.
A\c sa\u g\i daki her form\"ul i\c cin
serbest de\u gi\c skenleri ayn\i\ olan ve
$T_<{}^*$ teorisine g\"ore denk olan niceleyicisiz form\"ul\"u bulun.
\begin{enumerate}[a)]
\item 
$\Exists y(x<y)$,
\item 
$\Exists z(x<z\land z<y)$,
\item 
$\Exists z(x<z\land y<z)$,
\item
$\Exists y(x_0=y\land x_1<y\land x_1<x_2)$.
\end{enumerate}
\end{problem}

\begin{solution}\mbox{}
  \begin{enumerate}[a)]
    \item
$x=x$,
\item
$x<y$,
\item
$x=x\land y=y$,
\item
$x_1<x_0\land x_1<x_2$.
  \end{enumerate}
\end{solution}

\newpage

\begin{problem}
$\str A$ bir $(A,<)$ do\u grusal s\i ralamas\i,
$n\in\upomega$, ve $\vec a\in A^n$ olsun.
Her $i$ ve $j$ i\c cin, $i<j<n$ durumunda,
\begin{compactitem}
\item
  $a_i<a_j$ (yani $\str A\models a_i<a_j$) ise $\phi_{ij}$, $x_i<x_j$ olsun,
\item
  $a_i=a_j$ ise $\phi_{ij}$, $x_i=x_j$ olsun,
\item
  $a_j<a_i$ ise $\phi_{ij}$, $x_j<x_i$ olsun.
\end{compactitem}
Tan\i m\i m\i za g\"ore $n$-konumlu
\begin{equation*}
\bigwedge_{i<j<n}\phi_{ij}
\end{equation*}
form\"ul\"u,
$\vec a$ listesinin \textbf{s\i ralama tipidir.}
\"Orne\u gin $\str A=(\upomega,<)$ ve $n=3$ durumunda
$(0,1,1)$ listesinin s\i ralama tipi
\begin{equation*}
x_0<x_1\land x_0<x_2\land x_1=x_2,
\end{equation*}
ama $x_0<x_1\land x_2<x_0\land x_1=x_2$ form\"ul\"u, 
s\i ralama tipi de\u gildir
(\c c\"unk\"u bir s\i ralamada sa\u glanamaz).
\c Simdi $S_n$,
$n$-konumlu s\i ralama tiplerinin say\i s\i\ olsun.
\begin{enumerate}[a)]
\item 
Hesaplamalar\i n\i z\i\ g\"ostererek 
%a\c sa\u g\i daki
tabloyu doldurun.\hfill
\renewcommand{\arraystretch}{1.5}
  \begin{tabular}[t]{cc}
    $n$&$S_n$\\\hline
$2$&$3$\\\hline
$3$&\\\hline
$4$&\\\hline
  \end{tabular}
\item
A\c sa\u g\i daki denklemi, $x$ i\c cin \c c\"oz\"un.
\begin{equation*}
S_5=5!+\binom52\cdot4!+x+5\cdot\binom42\cdot3+\left(5+\binom52\right)\cdot2+1.
\end{equation*}
\end{enumerate}
\end{problem}

\begin{solution}
$x=\binom53\cdot3!=60$ ve
  \begin{gather*}
    S_3=3!+\binom32\cdot2!+1=6+6+1=13,\\
S_4=4!+\binom42\cdot3!+\binom43\cdot2+\binom42+ 1
   =24+      36       +      8       +   6    + 1 = 75.
  \end{gather*}
\end{solution}

\newpage

\begin{problem}
\.Imzas\i\ $\{E\}$ olan $T_{2,\infty}$ teorisinin her $\str A$ modeli i\c cin
$E^{\str A}$, iki s\i n\i fl\i\ denklik ba\u g\i nt\i s\i d\i r,
ve bu ba\u g\i nt\i n\i n her s\i n\i f sonsuzdur.
\begin{enumerate}[a)]
\item 
$T_{2,\infty}$ i\c cin aksiyomlar\i\ yaz\i n.
\item
A\c sa\u g\i daki her form\"ul i\c cin
serbest de\u gi\c skenleri ayn\i\ olan ve
$T_{2,\infty}$ teorisine g\"ore denk olan niceleyicisiz form\"ul\"u bulun.
\begin{enumerate}[i.]
\item 
$\Exists z\lnot(x\mathrel Ez\lor y\mathrel Ez)$,
\item
$\Exists y(x_0\mathrel Ey\land x_1\mathrel Ey\land\lnot\; x_2\mathrel Ey)$. 
\item
$\Exists y(x_0\mathrel Ey\land x_1\mathrel Ey\land\lnot\; x_2\mathrel Ey
\land x_0\neq y)$. 
\end{enumerate}
\end{enumerate}
\end{problem}

\begin{solution}
  \begin{asparaenum}[a)]
  \item 
Aksiyomlar,
\begin{gather*}
    \Forall xx\mathrel Ex,\\
\Forall x\Forall y(x\mathrel Ey\lto y\mathrel Ex),\\
\Forall x\Forall y\Forall z(x\mathrel Ey\land y\mathrel Ez\lto x\mathrel Ez),\\
\Forall x\Forall y\Forall z
(\lnot x\mathrel Ey\lto x\mathrel Ez\lor y\mathrel Ez),
  \end{gather*}
ve her $n$ do\u gal say\i s\i\ i\c cin
\begin{multline*}
  \Exists{x_0}\cdots\Exists{x_{2n-1}}\left(\lnot x_0\mathrel Ex_n\land
\bigwedge_{i<j<n}(x_i\neq x_j\land x_{n+i}\neq x_{n+j})\land{}\right.\\
\left.{}\land\bigwedge_{0<i<n}(x_0\mathrel Ex_i
\land x_n\mathrel Ex_{n+i})\right).
\end{multline*}
\item\mbox{}
  \begin{enumerate}[i.]
  \item 
$x\mathrel Ey$,
\item
$x_1\mathrel Ex_0\land\lnot\; x_2\mathrel Ex_0$,
\item
ayn\i: $x_1\mathrel Ex_0\land\lnot\; x_2\mathrel Ex_0$. 
  \end{enumerate}
  \end{asparaenum}
\end{solution}

\newpage

\begin{problem}
  \begin{enumerate}[a)]
\item
$\phi$ ve $\psi$ niceleyicisiz ise 
$\Exists x\phi\lto\Forall x\psi$ form\"ul\"un\"u
\"onekli bi\c cimde 
(yani niceleyicilerin \"onde oldu\u gu bi\c cimde) yaz\i n.
\item
  $T_{\Q}$, $\Q$ \"uzerinde do\u grusal uzaylar teorisi olsun.
$T_{\Q}\cup\{\sigma\}$ k\"umesinin tam bir teorinin aksiyom k\"umesi oldu\u gu
$\sigma$ c\"umlesini yaz\i n.
  \item 
Bo\c s imzada tam bir teorinin aksiyomlar\i\ yaz\i n.
\item
Bo\c s imzada ka\c c tane tam teori vard\i r?  K\i saca a\c c\i klay\i n.
  \end{enumerate}
\end{problem}

\begin{solution}\mbox{}
  \begin{enumerate}[a)]
    \item
$\Forall x\Forall y(\phi\lto\psi^x_y)$.
\item
$\Exists xx\neq0$.
\item
Her $n$ do\u gal say\i s\i\ i\c cin
\begin{equation*}
  \Exists{x_0}\cdots\Exists{x_n}\bigwedge_{i<j\leq n}x_i\neq x_j.
\end{equation*}
\item
Her $n$ i\c cin $n\in\upomega$ ise $T_n$,
$n$-elemanl\i\ k\"umeler teorisi olsun, 
ve $T_{\upomega}$, yukar\i daki teori olsun.
Bunlar bo\c s imzadaki tam teorilerdir \c c\"unk\"u
\begin{compactitem}
  \item
$A$ sonlu ise bir $n$ i\c cin $A\models T_n$;
\item
her $n$ i\c cin $T_n$ teorisinin modelleri izomorftur,
dolay\i s\i yla $T_n$ tam ve $T_n=\Th A$;
\item
niceleyicilerin giderilmesinden
$T_{\upomega}$ teorisinin tamli\u g\i n\i\ biliyoruz;
\item
$A$ sonsuz ise $A\models T_{\upomega}$, dolay\i s\i yla $\Th A=T_{\upomega}$.
\end{compactitem}
  \end{enumerate}
\end{solution}

\end{document}
